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 cs-401r:homework-0.1 [2014/09/03 06:00]ringger created cs-401r:homework-0.1 [2014/12/31 23:17] (current)ringger 2014/12/31 23:17 ringger 2014/09/03 06:00 ringger created 2014/12/31 23:17 ringger 2014/09/03 06:00 ringger created Line 17: Line 17: === Question 1: Useful theorems in probability theory === === Question 1: Useful theorems in probability theory === - [50 points; 10 per sub‐problem] + [50 points; 10 per sub-problem] (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. ​ Develop your proof first in terms of sets and then translate into probabilities;​ use set theoretic operations on sets and arithmetic operators on probabilities. ​ Be sure to apply [[Proofs|good proof technique]]:​ justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification,​ you must first satisfy the conditions / pre-requisites of the axiom. ​ Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. ​ Develop your proof first in terms of sets and then translate into probabilities;​ use set theoretic operations on sets and arithmetic operators on probabilities. ​ Be sure to apply [[Proofs|good proof technique]]:​ justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification,​ you must first satisfy the conditions / pre-requisites of the axiom. ​ - # <​math>​P(B - A) = P(B) - P(A \cap B)​ + # $P(B - A) = P(B) - P(A \cap B)$ - # <​math>​P(A \cup B) = P(A) + P(B) - P(A \cap B) ​(the addition rule) + # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) #* Hint: refer to part result #1 as a part of your proof of part #2 #* Hint: refer to part result #1 as a part of your proof of part #2 - # <​math>​P(\varnothing) = 0​ + # $P(\varnothing) = 0$ - # <​math>​P(\neg A) = 1 - P(A)​ + # $P(\neg A) = 1 - P(A)$ #* Hint: refer to part result #1 as a part of your proof of part #4 #* Hint: refer to part result #1 as a part of your proof of part #4 - # <​math>​A \subseteq B \Rightarrow P(A) \leq P(B)​ + # $A \subseteq B \Rightarrow P(A) \leq P(B)$ - #* Hint: let <​math>​C = B - A ​(set difference). + #* Hint: let $C = B - A$ (set difference). === Question 2: Joint Probability === === Question 2: Joint Probability === Line 39: Line 39: * Let <​tt>​is-abbreviation​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word​ denote "a period occurs after a three letter word" * Let <​tt>​is-abbreviation​ denote the event "this period indicates an abbreviation",​ and let <​tt>​three-letter-word​ denote "a period occurs after a three letter word" * Assume the following probabilities:​ * Assume the following probabilities:​ - ** <​math>​P(​<​tt>​is-abbreviation​<​math> ​| ​<​tt>​three-letter-word​<​math>​) = 0.8​ + ** $P($<​tt>​is-abbreviation​$|$<​tt>​three-letter-word​$) = 0.8$ - ** <​math>​P(​<​tt>​three-letter-word​<​math>​) = 0.0003​ + ** $P($<​tt>​three-letter-word​$) = 0.0003$ === Question 3: Independence === === Question 3: Independence === Line 47: Line 47: (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) (Adapted from: Manning & Schuetze, p. 59, exercise 2.4) - Are <​math>​X ​and <​math>​Y ​as defined in the following table independently distributed?​ + Are $X$ and $Y$ as defined in the following table independently distributed?​ ​ ​
<​math>​x$x$
<​math>​y$y$
<​math>​P(X=x, Y=y)$P(X=x, Y=y)$
- + <​td>​0​ <​td>​0​ <​td>​0​ <​td>​0​ Line 58: Line 58: - + <​td>​0​ <​td>​0​ <​td>​1​ <​td>​1​ Line 65: Line 65: - + <​td>​0.32​ <​td>​0.32​ <​td>​0.08​ <​td>​0.08​ Line 76: Line 76: [10 points] [10 points] - # How many possible ways can you completely factor the joint distribution ​<​math>​P(X_1, X_2, X_3, X_4, X_5, X_6)​? + # How many possible ways can you completely factor the joint distribution ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$? - # For some arbitrary joint probability distribution on six random variables ​<​math>​P(X_1, X_2, X_3, X_4, X_5, X_6)​, apply the chain rule to completely factor this distribution in one way. + # For some arbitrary joint probability distribution on six random variables ​$P(X_1, X_2, X_3, X_4, X_5, X_6)$, apply the chain rule to completely factor this distribution in one way. === Question 5: Conditional Probability === === Question 5: Conditional Probability === Line 87: Line 87: [10 points] [10 points] - Prove the chain rule, namely that <​math>​P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)​.  Justify each step.  Use the same standard of proof as in problem #1 above. + Prove the chain rule, namely that $P(\cap^n_{i=1} A_i) = P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdot ...\cdot P(A_n | \cap^{n-1}_{i=1} A_i)$.  Justify each step.  Use the same standard of proof as in problem #1 above. * Hint: you could use induction (but that isn't the only way). * Hint: you could use induction (but that isn't the only way).
cs-401r/homework-0.1.txt · Last modified: 2014/12/31 23:17 by ringger 